Question 13 of 14advanced🔍 AnalyzeLong Answer5 marks

How does the procedure of drawing a histogram differ when class intervals are unequal in comparison to equal class intervals in a frequency table?

Correct Answer

When class intervals are equal, drawing a histogram is straightforward. The rectangles are drawn with their heights directly proportional to the frequencies of the class intervals. Since all class intervals have the same width, the area of each rectangle automatically becomes proportional to its frequency.

However, when class intervals are unequal, the procedure changes significantly. The height of the rectangle cannot directly represent the frequency because the widths vary. Instead, we must calculate frequency density, which is the quotient of frequency and the width of the class interval.

The formula is: Frequency Density = Frequency ÷ Class Width

This adjustment ensures that the area of each rectangle remains proportional to the class frequency. In a histogram, the width is as important as its height. Unlike bar diagrams where only height matters, in histograms, the area represents the frequency.

For example, when tabulating deaths by age at death, shorter age intervals may be used at the beginning when death rates are high. Without adjusting for unequal widths, the histogram would misrepresent the actual distribution.

In both cases, the rectangles are drawn adjacent to each other with no space between them, as the lower class boundary of one interval fuses with the upper class boundary of the previous interval.

Exercise: EXERCISES | Q: 13 | (Page Page 18)
For More Understanding

Explanation

The textbook clearly explains that for equal class intervals, the area of rectangles is directly proportional to frequencies since widths are uniform. However, for unequal class intervals, we must adjust the height using frequency density (frequency ÷ width) to maintain the proportionality of area to frequency. This is crucial because in histograms, unlike bar diagrams, width matters as much as height. The context provides the example of deaths by age where varying widths are used for meaningful representation.

Solution Steps

  1. Step 1: Identify whether class intervals are equal or unequal

  2. Step 2: For equal intervals, plot height = frequency directly

  3. Step 3: For unequal intervals, calculate frequency density = frequency ÷ class width

  4. Step 4: Plot rectangles with height = frequency density for unequal intervals

  5. Step 5: Ensure rectangles are adjacent with no gaps in both cases

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